low energy properties of the random displacement model

Journal of Functional Analysis 256 (2009) 2725–2740

www.elsevier.com/locate/jfa

Low energy properties of the random displacementmodel

Jeff Baker a, Michael Loss b, Günter Stolz c,∗

a Southern Company Generation, 600 North 18th Street, Birmingham, AL 35291-2641, USAb Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, USA

c University of Alabama at Birmingham, Department of Mathematics, Birmingham, AL 35294-1170, USA

Received 19 August 2008; accepted 23 January 2009

Available online 31 January 2009

Communicated by D. Stroock

Abstract

We study low-energy properties of the random displacement model, a random Schrödinger operatordescribing an electron in a randomly deformed lattice. All periodic displacement configurations whichminimize the bottom of the spectrum are characterized. While this configuration is essentially uniquefor dimension greater than one, there are infinitely many different minimizing configurations in the one-dimensional case. The latter leads to unusual low energy asymptotics for the integrated density of states ofthe one-dimensional random displacement model. For symmetric Bernoulli-distributed displacements it hasa 1/ log2-singularity at the bottom of the spectrum. In particular, it is not Hölder-continuous.© 2009 Elsevier Inc. All rights reserved.

Keywords: Random Schrödinger operator; Random displacement model; Integrated density of states

1. Introduction

We consider the so-called random displacement model, i.e. the random Schrödinger operator

Hω = −� + Vω, (1)

* Corresponding author. Fax: +1 205 934 9025.E-mail addresses: [email protected] (J. Baker), [email protected] (M. Loss), [email protected]

(G. Stolz).

0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jfa.2009.01.022

2726 J. Baker et al. / Journal of Functional Analysis 256 (2009) 2725–2740

where the random potential Vω is given by displacing a single site potential q from the pointsof Z

d ,

Vω(x) =∑i∈Zd

q(x − i − ωi). (2)

For the real-valued single site potential q we assume q ∈ L∞(Rd) and suppq ⊂ [−r, r]d forsome r < 1/2. We also assume that q is reflection symmetric at each coordinate hyperplane,i.e. symmetric in each variable with the remaining variables fixed. Throughout this paper wewill consider displacement configurations ω = (ωi)i∈Zd such that ωi ∈ [−dmax, dmax]d for all i,where r + dmax = 1/2. The latter ensures that the displaced single site potentials in (2) do notoverlap.

While the random displacement model is a physically quite natural way to describe structuraldisorder in a solid, it is mathematically much less well understood than Anderson-type mod-els, where the disorder enters in the form of random couplings at the single-site potentials. Thisis mostly due to the fact that the random displacement model depends non-monotonously onthe random parameters (in quadratic form sense). Anderson-type models, on the other hand, aremonotonous in this sense, at least if the single-site potentials have fixed sign. The consequentialchallenge in determining the spectral properties of the random displacement model, in particularthe low energy behavior, lies in having to gain a deeper understanding of the interaction betweenmultiple random parameters as well as the interplay between kinetic and potential energy. Spec-tral averaging arguments using individual random parameters, a common tool in the theory ofAnderson models, are not available here.

This is one of the main reasons why it is not yet known if the multi-dimensional random dis-placement model is localized at the bottom of the spectrum (in d = 1 localization at all energiesfollows from the results in [6], see also [18]). The only known result on localization for the multi-dimensional random displacement model is due to Klopp [10], who considered the semi-classicalversion −h2�+Vω of the random displacement model and identified a localized region near thebottom of the spectrum for sufficiently small values of the semi-classical parameter h.

An attempt to understand the low energy properties of the random displacement model hasto start with describing the mechanism which characterizes the bottom of the spectrum. Underthe above assumptions this has been achieved in [2] by identifying the periodic displacementconfiguration ωmin, see (4) below, which leads to the minimum of the almost sure spectrumof Hω .

Here we continue our study of the low energy properties of the random displacement modelby first characterizing the set of all minimizing periodic configurations. It turns out that ωmin, upto trivial translations, is the unique minimizer in d � 2, while in d = 1 there are infinitely manyperiodic minimizers, which can be explicitly characterized. These results are stated in Section 2and proven in Section 3.

We then move to studying the low energy asymptotics of the integrated density of states(IDS) for the random displacement model. Showing smallness of the IDS near the bottom ofthe spectrum, typically in the form of Lifshits tails, is an important step in all approaches tolow-energy localization for multi-dimensional random Schrödinger operators. It is interpreted asshowing that the bottom of the spectrum is a fluctuation boundary.

However, as we will show here, for the one-dimensional random displacement model thebehavior of the IDS can be very different. If the displacements only take values dmax or −dmax,

J. Baker et al. / Journal of Functional Analysis 256 (2009) 2725–2740 2727

both with equal probability, then the IDS N(E) has a very strong singularity at the bottom of thespectrum,

N(E) � C

log2(E − E0)(3)

for E ∈ (E0,E0 + ε) and constants C > 0 and ε > 0, see Theorem 4.1. Here E0 denotes thealmost sure minimum of the spectrum of Hω . Thus the IDS is not even Hölder-continuous at E0,a new phenomenon which to our knowledge has not been found for any other models of ran-dom operators. This and related results for the IDS of the one-dimensional random displacementmodel with other distributions of the displacements are proven in Section 4. For example, theextreme behavior (3) only appears for the given case of a symmetric Bernoulli distribution of thedisplacements, see Theorem 4.2. But, in d = 1 and as long as the distribution of the displace-ments ωn is symmetric, one never gets Lifshits tails (Theorem 4.3).

In Section 5 we comment on several open problems, in particular on our expectation thatthe uniqueness of the periodic minimizing configuration in d � 2 should indicate different lowenergy asymptotics for the IDS than in d = 1 (e.g. the appearance of some form of Lifshits tails).We will also discuss related recent works by Klopp and Nakamura [11] and Fukushima [8].

2. Periodic configurations which minimize the ground state energy

In [2] we have identified a simple periodic configuration of displacements which leads to thelowest possible spectral minimum for Hω among all configurations ω:

Proposition 2.1. (See Theorem 1.1 in [2].) Let ωmin be given by

ωmini = (

(−1)i1dmax, . . . , (−1)id dmax)

(4)

for all i = (i1, . . . , id ) ∈ Zd . Then

E0 = minσ(Hωmin), (5)

where E0 := infω minσ(Hω).

This configuration is 2-periodic in each coordinate, where in each period cell 2d single sitescluster together in adjacent corners of unit cubes, see Fig. 1 for d = 2.

Here our first goal is to characterize all periodic configurations ω such that minσ(Hω) = E0.For this we will also use another result found in [2]: Let q be as above, a ∈ [−dmax, dmax]d ,Λ0 = (− 1

2 , 12 )d the unit cube centered at 0, and HN

Λ0(a) = −� + q(x − a) on L2(Λ0) with

Neumann boundary condition on ∂Λ0. By S we denote the connected component of Λ0 \ suppq

containing the boundary of Λ0.

Proposition 2.2. (See Theorem 1.3 of [2].) If

E0(a) := minσ(HN

Λ0(a)

), (6)

then the following alternative holds: Either


Fig. 1. The support of Vωmin for d = 2.

(i) E0(a) is strictly maximized at a = 0 and strictly minimized in the 2d corners(±dmax, . . . ,±dmax) of [−dmax, dmax]d , or

(ii) E0(a) is identically zero. In this case the corresponding eigenfunction is constant on S.

Note that the way in which alternative (ii) is phrased clarifies how this was stated in [2] tocorrectly reflect what is proven there.

Alternative (i) holds, for example, if q is non-zero and sign-definite, in which case E0(a)

never vanishes. But sign-definiteness of q is far from necessary for alternative (i). In fact, it iseasy to see that for every r < 1/2 the set of all q with support in [−r, r]d for E0(0) �= 0 is openand dense in L∞([−r, r]d). Thus alternative (i) is generic.

But alternative (ii) holds in non-trivial cases (where q �= 0). To construct an example, let0 < s < r < 1/2 and let q0(x) = 1 for |x| � s and q0(x) = 0 for |x| > s. The Neumann problemfor −� + q0 on L2(|x| � r) has lowest eigenvalue 0 < E0 < 1. Choose

q(x) =⎧⎨⎩

1 − E0 for |x| � s,

−E0 for s < |x| � r ,

0 for |x| > r ,(7)

which is radially symmetric and thus reflection symmetric in each variable. The Neumann prob-lem for −� + q on L2(|x| � r) has lowest eigenvalue 0. The corresponding eigenfunction ϕ

satisfies Neumann conditions on |x| = r and is radially symmetric. Therefore it is constant on|x| = r and can be extended by that constant to Λ0, giving the ground state of the Neumannproblem for −� + q on L2(Λ0). By translation one gets the ground state of HN

Λ0(a) for every a,

i.e. E0(a) ≡ 0.

Theorem 2.3.

(a) If alternative (ii) holds, then minσ(Hω) = 0 for all configurations ω = (ωi)i∈Zd , with ωi ∈[−dmax, dmax]d for all i.

(b) If alternative (i) holds, d � 2 and r < 1/4, then ωmin given by (4) is, up to translations, theunique periodic configuration with minσ(Hωmin) = E0.


We do not believe that the extra condition r < 1/4 (beyond r < 1/2) is required in Theo-rem 2.3, but we need it in our proof.

It remains to settle the case of alternative (i) and d = 1. In this case the periodic minimizer ishighly non-unique, but the set of all periodic minimizers can be characterized by our next result.Here, for L ∈ N let SL denote the set of all L-periodic configurations (ωi)i∈Z such that ωi =−dmax or ωi = dmax for all i. Furthermore, for ω ∈ SL let n±(ω) be the number of i ∈ {1, . . . ,L}with ωi = ±dmax.

Theorem 2.4. Let d = 1 and q such that alternative (i) holds. An L-periodic configuration ω

satisfies minσ(Hω) = E0 if and only if

L is even, ω ∈ SL, and n−(ω) = n+(ω). (8)

Thus, in each period interval of Vω, equally many of the single site potentials sit at the extremeright and the extreme left of their allowed range of positions. The dimer configuration ωmin ismerely a special case of this situation, namely the only configuration satisfying (8) for L = 2.For L = 4 a different configuration is given by choosing the signs of ω1, . . . ,ω4 ∈ {±dmax} as+ + −−. For L = 6 two more configurations satisfy (8): + + + − −− and + + − + −− (withothers being equivalent to these via translation and inversion). The number of configurationssatisfying (8) grows rapidly with L, in fact exponentially.

The short explanation behind Theorems 2.3 and 2.4 is that for d = 1 the ground states ofHN

Λ0(a) for extremal positions of a can be made to match after suitable re-scaling, while the

richer geometry for d � 2 prevents this for all configurations other than ωmin.

3. Proof of Theorems 2.3 and 2.4

Part (a) of Theorem 2.3 easily follows from known facts: Under alternative (ii) the groundstate energy of the Neumann problem for −� + q restricted to (−r, r)d is 0. Let ϕ be thecorresponding positive normalized eigenfunction. For any given configuration ω place a trans-late of ϕ at i + ωi for each i ∈ Z

d and extend by a constant to the exterior of the convexhull of suppq(· − i − ωi). This results in a globally bounded, positive weak solution ϕ̃ of−�ϕ̃ + Vωϕ̃ = 0. By Shnol’s theorem we have 0 ∈ σ(Hω). As ϕ̃ is positive it also followsfrom Theorem C.8.1 in [14] that 0 � minσ(Hω). Thus minσ(Hω) = 0 for all configurations ω.

For the remainder of this section we will assume that alternative (i) holds. If ω is a pe-riodic configuration with period L = (L1, . . . ,Ld) ∈ N

d , i.e. such that ωi+(n1L1,...,ndLd) = ωi

for all i ∈ Zd and (n1, . . . , nd) ∈ Z

d , we choose Λ = (1/2, . . . ,L1 + 1/2) · · · × · · · (1/2, . . . ,

Ld + 1/2) ⊂ Rd as period cell of the potential Vω given by (2). By HP

ω,Λ and HNω,Λ we de-

note the restrictions of −� + Vω to L2(Λ) with periodic and Neumann boundary conditions,respectively, and denote their lowest eigenvalues by E0(H

Pω,Λ) and E0(H

Nω,Λ).

The proofs of Theorems 2.3(b) and Theorem 2.4 will be based on the following result whichholds for arbitrary dimension. It shows, in particular, that in minimizing periodic configurationsall single site potentials necessarily must sit in the corners of unit cubes centered at the pointsof Z

d .

Lemma 3.1. Let ω be a periodic configuration with minσ(Hω) = E0. Then, for all i ∈ Zd ,

ωi ∈ {(a1, . . . , ad) ∈ Rd : ak ∈ {−dmax, dmax} for all k = 1, . . . , d}. Moreover, in this case

E0(HPω,Λ) = E0(H

Nω,Λ) and the ground state eigenfunction ψω of HN

ω,Λ satisfies Neumann

boundary conditions on the boundary of each unit cube Λi centered at i ∈ Λ ∩ Zd .


Proof. By assumption and Floquet–Bloch theory E0 = minσ(Hω) = E0(HPω,Λ). Also, by the

variational principle, E0(HPω,Λ) � E0(H

Nω,Λ). The ground state ψω minimizes the quadratic form

of HNω,Λ, thus

E0(HN

ω,Λ

) =∫Λ

|∇ψω|2 + ∫Λ

∑i∈Λ∩Zd q(x − i − ωi)|ψω|2∫Λ

|ψω|2

=∑

i∈Λ∩Zd

∫Λi

|∇ψω|2 + ∫Λi

q(x − i − ωi)|ψω|2∫Λi

|ψω|2 ·∫Λi

|ψω|2∫Λ

|ψω|2

�∑

i∈Λ∩Zd

E0(ωi)

∫Λi

|ψω|2∫Λ

|ψω|2 �∑

i∈Λ∩Zd

E0

∫Λi

|ψω|2∫Λ

|ψω|2 = E0, (9)

where E0(ωi) is given by (6). In the second to last inequality we have used the variational prin-ciple as well as the fact that ψω does not vanish on any of the Λi .

We conclude that all inequalities above must indeed be equalities, which immediately givesE0(H

Pω,Λ) = E0(H

Nω,Λ). If for at least one ωi /∈ {a: ak ∈ {−dmax, dmax}, k = 1, . . . , d}, then the

last inequality in (9) would be strict, given that we are in alternative (i). We conclude that all ωi

sit in a corner. Finally, we see that

∫Λi

|∇ψω|2 + ∫Λi

q(x − i − ωi)|ψω|2∫Λi

|ψω|2 = E0(ωi) (10)

for each i. Thus the restriction of ψω to Λi is the ground state for the Neumann problem of−� + V (x − i − ωi) on Λi and thus satisfies Neumann boundary conditions on each Λi . �

We now consider d = 1, still under alternative (i), where we use one more lemma to preparefor the proof of Theorem 2.4.

Lemma 3.2. Let d = 1 and HNΛ0

(a) be the restriction of −d2/dx2 + q(x − a) to L2(− 12 , 1

2 ) with

Neumann boundary conditions. Let ψ be the positive normalized ground state of HNΛ0

(dmax).

Then ψ( 12 ) �= ψ(− 1

2 ).

Proof. Suppose that ψ( 12 ) = ψ(− 1

2 ), then ψ coincides with the periodic ground state of−d2/dx2 + q(x − dmax) on L2(− 1

2 , 12 ). Due to the symmetry of q and the uniqueness of the

periodic ground state, it must therefore also satisfy ψ ′(− 12 + dmax) = 0 (considering (− 1

2 , 12 ) as

a 1-torus, − 12 + dmax lies opposite to dmax). As q(x) = 0 for x ∈ (− 1

2 ,− 12 + dmax), we have that

−ψ ′′ = E0(dmax)ψ on [− 12 ,− 1

2 + dmax] and

ψ ′(

−1

2

)= ψ ′

(−1

2+ dmax

)= 0. (11)

If E0(dmax) < 0, then ψ ′′ = −E0(dmax)ψ > 0 and thus ψ is strictly convex on [−1/2,

−1/2 + dmax], contradicting (11). Similarly, E0(dmax) > 0 would yield strict concavity of ψ ,


again contradicting (11). Thus E0(dmax) = 0 and it follows from (11) that ψ must be constantoutside the support of q(x − dmax). This contradicts that we have assumed alternative (i). �

We can now complete the proof of Theorem 2.4: Let ω be an L-periodic configuration whichsatisfies minσ(Hω) = E0. By Lemma 3.1 we have ωi ∈ {±dmax} for all i = 1, . . . ,L, and alsoE0 = minσ(HP

ω,Λ) = minσ(HNω,Λ), where Λ = ( 1

2 ,L + 12 ).

Let uD and uN be the solutions of −u′′ + V u = E0u which satisfy uD( 12 ) = 0, u′

D( 12 ) = 1,

uN( 12 ) = 1, u′

N( 12 ) = 0. E0 is a Neumann eigenvalue on Λ and thus u′

N(L+ 12 ) = 0. The transfer

matrix for Hω at E0 from 1/2 to L + 1/2 is given by

T =(

uN(L + 12 ) uD(L + 1

2 )

u′N(L + 1

2 ) u′D(L + 1

2 )

). (12)

This implies

1 = detT = uN(L + 1/2)u′D(L + 1/2). (13)

Moreover, as E0 is also an eigenvalue for periodic boundary conditions,

2 = trT = uN(L + 1/2) + u′D(L + 1/2). (14)

We conclude from (13) and (14) that

uN(L + 1/2) = 1 = u′D(1/2), (15)

meaning that uN is both the Neumann and periodic eigenfunction to E0.We can use Lemma 3.2 to understand the detailed structure of uN : Let ψ be the normalized

ground state of HNΛ0

(dmax) as given there. Then, by symmetry of q , the normalized ground state

of HNΛ0

(−dmax) is given by ψ̃(x) = ψ(−x). As ωi ∈ {±dmax} for all i = 1, . . . ,L, we can con-

struct uN by concatenating suitably re-scaled versions of ψ and ψ̃ , respectively, on the intervals[i − 1

2 , i + 12 ]. With the positive number r = ψ( 1

2 )/ψ(− 12 ) �= 1 from Lemma 3.2 we thus have

uN(i + 1/2)/uN(i − 1/2) ={

r if ωi = dmax,1r

if ωi = −dmax.(16)

The accumulative effect is that uN(L + 12 ) = rn+(ω)−n−(ω)uN( 1

2 ). We conclude from (15) thatn+(ω) = n−(ω), in particular that L is even, which completes the proof of Theorem 2.4.

We now start with preparations for the proof of Theorem 2.3(b).

Lemma 3.3. Consider a connected open region D in Rd , d � 2 and a hyperplane P that divides

this region into two nonempty subregions. Denote by σ the reflection about P and assume thatD ∩ σ(D) is connected. Let E ∈ R and, in D, let u be a solution of the equation

−�u = Eu (17)

which satisfies the condition ∂u∂n

= 0 on P ∩ D. Then u can be extended to a symmetric functionw on D ∪ σ(D) which satisfies the equation −�u = Eu in this region.


Proof. Pick a point x0 ∈ P ∩ D, which we may assume to be the origin. Pick a ball B ⊂ D

centered at the origin and pick coordinates x1, . . . , xn so that x′ = (x1, . . . , xn−1) are coordinatesin P and xn is the coordinate normal to P . Consider the function

v(x′, xn) = u(x′, xn) − u(x′,−xn)

which satisfies (17) and vanishes on B ∩ P identically. Its first normal derivative satisfies

∂v

∂xn

(x′,0) = 0. (18)

Since

0 = Ev(x′,0) = −�v(x′,0) = − ∂2v

∂x2n

(x′,0) (19)

we also have that

∂2v

∂x2n

(x′,0) = 0. (20)

Further, since

0 = E

(∂

∂xn

v

)(x′,0) = −

(�

∂

∂xn

v

)(x′,0) (21)

= −(

∂3

∂x3n

v

)(x′,0) −

(�′ ∂

∂xn

v

)(x′,0) (22)

we obtain from (18) that

∂3v

∂x3n

(x′,0) = 0. (23)

Continuing in this fashion we deduce that

∂kv

∂xkn

(x′,0) = 0, (24)

for k = 0,1,2, . . . . In particular, this implies that all derivatives of v vanish at the origin. Since v

solves (17) it is a real analytic function and hence it vanishes in the ball B . Since x0 ∈ D ∩ σ(D)

and D ∩ σ(D) is connected we learn that v vanishes everywhere in D ∩ σ(D). Thus u is sym-metric in D ∩ σ(D) with respect to reflection about the plane P . Next we prolong u to thecomplement of D ∩ σ(D) in D ∪ σ(D) by setting

w(x) ={

u(x), x ∈ D,(25)

u(σ (x)), x ∈ σ(D).


Note that this function is defined since u(x) and u(σ (x)) coincide on D ∩ σ(D). Moreover,x ∈ D ∪ σ(D) means that x ∈ D or x ∈ σ(D) or both. In any case, by assumption and the factthat the Laplace operator commutes with reflections, w(x) satisfies Eq. (17) at this point whichproves the lemma. �

Given the previous lemma, we can now complete the proof of Theorem 2.3(b). Suppose thatd � 2 and ω is a periodic configuration with minσ(Hω) = E0, but not a translate of ωmin. Thenthere must be two adjacent unit cubes, say Λ and Λ′, such that the potential on the union R =Λ ∪ Λ′ of these cubes is not symmetric with respect to reflection about their common face. Thiscommon face defines a hyperplane P . By Lemma 3.1, the two potential sites of Vω restricted toR are supported in corners of Λ and Λ′, respectively, in the sense that [a1 − r, a1 + r] × · · · ×[ad − r, ad + r] sits in a corner of Λ, and similar for a′ and Λ′. Also, the positive ground stateeigenfunction ψω of Hω satisfies Neumann conditions on ∂R as well as on P . Let

D = R \ ([a1 − r, a1 + r] × · · · × [ad − r, ad + r] ∪ [a′1 − r, a′

1 + r] × · · ·× [a′

d − r, a′d + r]) (26)

and u the restriction of ψω to D. For these choices of D and u and E = E0 we can applyLemma 3.3. In particular, the assumption r < 1/4 assures the required connectedness of D andD ∩σ(D). Non-symmetry of a and a′ and again r < 1/4 implies that D ∪σ(D) is all of R. Thus,by Lemma 3.3, u can be extended to a symmetric function w on R which satisfies −�w = E0w

on R. Therefore it is the ground state for the Neumann problem of −� on R. This implies E0 = 0and that w is constant, a contradiction to the assumption of alternative (i).

4. Consequences for the integrated density of states

We now consider the random displacement model, i.e. the model (1), (2) for the case whereω = (ωi)i∈Zd is an array of i.i.d. vector-valued random variables with common distribution μ

supported on [−dmax, dmax]d . Here, as usual, we define

suppμ := {a ∈ R

d : μ({

x: |x − a| < ε})

> 0 for all ε > 0}.

Then the random operator Hω is ergodic with respect to translations in Zd and thus has all the

basic properties of ergodic operators, see e.g. [3]. In particular, the integrated density of states(IDS)

N(E) = limL→∞

1

|ΛL|E(trχ(−∞,E]

(HX

ω,ΛL

))(27)

exists for all energies E ∈ R. Here ΛL = ( 12 ,L + 1

2 )d and HXω,ΛL

is the restriction of Hω to

L2(ΛL) with boundary condition X ∈ {P,N,D}, as periodic (P), Neumann (N) and Dirichlet (D)boundary conditions all give the same limit in (27).


The spectrum σ(Hω) is almost surely deterministic, i.e. Σ = σ(Hω) for almost every ω, andgiven by the growth points of the non-decreasing function N(E). It can be characterized in termsof the spectra of those Hω for which the configuration ω is periodic,

Σ =⋃ω

σ(Hω), (28)

where the union is taken over all periodic ω such that ωi ∈ suppμ for all i. This correspondsto a well-known result for the Anderson model, e.g. [3], and is found with the same proof.If the support of the distribution μ contains all the corners {(±dmax, . . . ,±dmax)} of the cube[−dmax, dmax]d , then it follows from (28) and Proposition 2.1 that

minΣ = E0 = minσ(Hωmin).

For large classes of random Schrödinger operators it is known that the IDS vanishes rapidlyat the bottom of the spectrum E0, for example one has Lifshits tail behavior

N(E) ∼ e−c|E−E0|−d/2(29)

for Anderson models with sign-definite single-site potential, see e.g. [9,13] or [15] for properstatements, proofs and references to the original literature.

It turns out that for the random displacement model the behavior of the IDS at the bottom ofthe spectrum is much more subtle. Here we will present several results for the one-dimensionaldisplacement model, which were obtained in [1]. We will generally assume that alternative (i)holds. We expect quite different phenomena to appear for the multi-dimensional displacementmodel as well as for the case that the single site potential satisfies alternative (ii). More discussionof this is included in Section 5.

In the first result we will consider the one-dimensional Bernoulli displacement model, i.e. thecase where the distribution of the displacements ωi is given by

μ = 1

2δdmax + 1

2δ−dmax . (30)

It turns out that in this case the low-energy asymptotics of the IDS is at the opposite extreme ofLifshits tails:

Theorem 4.1. Let Hω be the one-dimensional symmetric Bernoulli displacement model given by(1), (2) and (30) and assume that alternative (i) holds. Then there exist C > 0 and ε > 0 suchthat

N(E) � C

log2(E − E0)(31)

for E ∈ (E0,E0 + ε).

As N(E0) = 0, this means that N(E) has infinite upper derivative at E = E0, i.e. the densityof states n(E) = N ′(E) has a strong singularity at the bottom of the spectrum. This is opposedto the case of Lifshits tails which would yield n(E0) = 0. In fact, (31) says that the IDS is not


even Hölder-continuous at E = E0, an even stronger singularity than one gets for the LaplacianH0 = −d2/dx2, where the IDS has a van Hove singularity C|E|1/2. For general one-dimensionalergodic Schrödinger operators (and for discrete ergodic Schrödinger operators also in higherdimension) the IDS is log-Hölder-continuous at all energies, i.e.

∣∣N(E) − N(E′)∣∣ � C

| log |E − E′|| (32)

for E close to E′, see [4,5]. Craig and Simon constructed examples of quasi-periodic potentialswhich show that the bound (32) is optimal. As far as we know, the result in Theorem 4.1 providesthe first known example of a random potential (with finite correlation length) where for at leastone energy the IDS is not Hölder-continuous and, in fact, close to the minimal possible regularityfor ergodic operators given by (32).

Proof. We will use the standard lower bound, e.g. [3],

N(E) � 1

LP(E0(H

Dω,L) < E

), (33)

which holds for arbitrary L, to be chosen later depending on E. Here HDω,L is short for HD

ω,ΛL,

ΛL = (1/2,L + 1/2).To show that E0(H

Dω,L) < E we will find ψω ∈ D(HD

ω,L) with ‖ψω‖ = 1 and

〈ψω,HDω,Lψω〉 < E. To construct ψω, let displacements ω = (ω1, . . . ,ωL) be given and let uN

be the solution of −u′′ + Vωu = E0u with uN( 12 ) = 1, u′

N( 12 ) = 0. Choose cut-off functions

θL ∈ C∞0 (R) with 0 � θL � 1, supp θL ⊂ [1,L], θL(x) = 1 for 3/2 � x � L − 1/2, and ‖θ ′

L‖∞and ‖θ ′′

L‖∞ uniformly bounded in L.As the ωi have distribution (30), we have ωi ∈ {±dmax} for all i. Thus the restriction of

−d2/dx2 + Vω to (i − 1/2, i + 1/2) has Neumann ground state energy E0 for all i ∈ {1, . . . ,L},which implies that

u′N(i + 1/2) = 0 for all i ∈ {1, . . . ,L}. (34)

We choose ψω := θLuN/‖θLuN‖ and calculate

⟨ψω,HD

ω,Lψω

⟩ − E0 = 〈θLuN,−θ ′′LuN 〉 − 2〈θLuN, θ ′

Lu′N 〉

‖θLuN‖2

�β̃(1 + u2

N(L + 1/2))∫ L−1/23/2 u2

N(x)dx

�β(1 + u2

N(L + 1/2))∑Li=1 u2

N(i + 1/2), (35)

where β̃ > 0 and β > 0 can be chosen uniformly in ω and L. Here we have repeatedly usedstandard a priori upper and lower bounds on solutions of −u′′ + V u = Eu, for example thatr(x) ∼ r(x + 1) and

∫ x+1x

u2 ∼ r2(x), where r(x) = (u2(x) + u′2(x))1/2 is the Prüfer amplitudeof u and constants can be chosen uniform as long as E and ‖V ‖∞ vary in a bounded interval, see


e.g. [16,17] for more details. We also use that by (34) the Prüfer amplitude of uN at the pointsi + 1/2 coincides with uN(i + 1/2).

Thus

P(E0

(HD

ω,L

)< E

)� P

(β

1 + u2N(L + 1/2)∑L

i=1 u2N(i + 1/2)

< E − E0

). (36)

Another consequence of ωi ∈ {±dmax} is that uN satisfies (16) for every i with a positive r �= 1,using that we are in alternative (i). Assume without restriction that r > 1 (if 0 < r < 1 then we cando the following construction from “right to left,” choosing uN(L+ 1/2) = 1, u′

N(L+ 1/2) = 0)and set

Xi = log(uN(i + 1/2)/uN(i − 1/2))

log r, (37)

i = 1, . . . ,L. The Xi are independent symmetric Bernoulli random variables with values ±1,and

u2N(i + 1/2) = e2Si log r , (38)

where Si = X1 + · · · + Xi . If Y := maxi=1,...,L Si , then it is a consequence of the reflectionprinciple for symmetric random walks, e.g. [7] that

P(Y �

√L

∣∣ SL � 0) = P

(SL � 2

√L

). (39)

The latter converges to π−1/2∫ ∞

2 exp(−y2/2) dy > 0 as L → ∞ by the central limit theorem.

Let AL := {ω | Y �√

L and SL � 0}. If Y �√

L, then∑L

i=1 u2N(L+1/2) � exp(2

√L log r).

Also, SL � 0 means u2N(L + 1/2) � 1. Thus (36) implies

P(E0

(HD

ω,L

)< E

)� P

(β

1 + u2N(L + 1/2)∑L

i=1 u2N(i + 1/2)

< E − E0 | AL

)P(AL)

= P(AL) � c0 > 0 (40)

if 2β exp(−2√

L log r) < E − E0 and L sufficiently large. This determines the choice of L ∈ N

for given E such that

1

2βe−2

√L−1 log r � E − E0 � 1

2βe−2

√L log r . (41)

Thus L ∼ (log 2β(E−E0)

log r)2. From (33) and (40) we have N(E) � c0/L, which, for E − E0 suffi-

ciently small, takes the form (31). �As mentioned above, (31) says in particular that the IDS is not Hölder-continuous at E = E0.

This is only possible if the distribution μ is concentrated in the extreme points dmax and −dmax,as is demonstrated by our next result.


Theorem 4.2. Suppose that the distribution μ of the ωi in the one-dimensional displacementmodel (1), (2) satisfies

μ((−dmax, dmax)

)> 0. (42)

Then the IDS N(E) is Hölder-continuous at E = E0.

This result may not be optimal. We expect that under the conditions of Theorem 4.2 one canat show that N(E) � Cα|E −E0|α near E0 for arbitrary α > 0. But, as long as the distribution μ

is chosen symmetric and not too small at ±dmax, one does not get Lifshits tail decay as in (29).To make this precise, define the Lifshits exponent γ at E0 by

γ = limE↓E0

log(− logN(E))

log(E − E0)(43)

whenever this limit exists. Note that γ � 0. If γ < 0, then it determines the asymptotics of theIDS in the sense that, up to logarithmic corrections, N(E) ∼ C1 exp(−C2(E −E0)

γ ) as E ↓ E0.

Theorem 4.3. Assume that the distribution μ is symmetric and satisfies

μ([dmax − ε, dmax] ∪ [−dmax,−dmax + ε]) � C1ε

N (44)

for some positive C1 and N and all ε > 0. Also assume that the single-site potential q is uni-formly Hölder-continuous, i.e. that |q(x) − q(y)| � C2|x − y|ρ for some C2 and ρ > 0 and allx, y.

Then γ = 0.

In the following we sketch the proofs of Theorems 4.2 and 4.3, referring for additional detailsto [1].

To prove Theorem 4.3 we follow the general strategy of the proof of Theorem 4.1, startingwith (33) and using the test function ψω := θLuN/‖θLuN‖. However, the construction of uN

needs to be modified as follows: On each interval [i − 1/2, i + 1/2], i ∈ {1, . . . ,L}, uN is chosento coincide with a constant multiple of the positive ground state of the Neumann problem for−d2/dx2 +q(x− i−ωi) on [i−1/2, i+1/2]. Scaling constants are chosen such that uN(1/2) =1 and uN is continuously differentiable throughout [1/2,L + 1/2]. As we now generally haveE0(ωi) �= E0, this leads to extra terms in the bound

N(E) � 1

LP

( |〈θLψω,−θ ′′Lψω〉| + 2|〈θLψω, θ ′

Lψ ′ω〉|

‖θLψω‖2

+L∑

i=1

(E0(ωi) − E0)‖θLψω‖2

‖θLψω‖2< E − E0

), (45)

here ‖θLψω‖21 := ∫ i

i−1 θ2Lψ2

ω.Due to the symmetry of μ, the numbers loguN(i + 1/2), i = 1, . . . ,L, are still a symmetric

random walk (but not Bernoulli). Versions of the reflection principle and central limit theorem for


general symmetric random walks and a choice of L as in (41) (with a suitable positive constantreplacing log r) lead to the bound

N(E) � C

LP

(L∑

i=1

(E0(ωi) − E0)‖θLψω‖2

i

‖θLψω‖2< E − E0

)

� C

LP

(L∑

i=1

(E0(ωi) − E0) < E − E0

)

� C

L

(P

(E0(ω1) − E0 <

E − E0

L

))L

. (46)

In Lemma 2.1 of [2] the continuity of E0(·) was shown. The proof given there provides the bound

∣∣E0(a1) − E0(a2)∣∣p � C

∫ ∣∣q(x − a1) − q(x − a2)∣∣p dx

for any p � 2. Uniform Hölder continuity of q gives Hölder continuity of E0(·). Using E0 =E0(dmax) = E0(−dmax) and (44) we see that P(E0(ω1) − E0 < δ) � C1(δ/C)N/ρ . We pluginto (46)

N(E) � C

L

(E − E0

L

)N/ρ

.

From this bound, having chosen L through (41), a calculation shows that the Lifshits exponentvanishes.

The proof of Theorem 4.2 is based on the standard upper bound, e.g. [3],

N(E) � CP(E0

(HN

ω,L

)� E

). (47)

We choose L through

s0e−2C1(L+1) � E − E0 � s0e

−2C1L (48)

with constants s0 and C1 to be determined later. By the calculation done in (9),

E0(HN

ω,L

)�

L∑i=1

E0(ωi)

∫ i

i−1 |ψω|2∫ L+1/21/2 |ψω|2

, (49)

where ψω is the ground state of HNω,L. By a priori bounds (e.g. [16]) there exists C1 > 0 such

that

e−C1L �i∫

|ψω|2 dx � eC1L

i−1


uniformly in L ∈ N, i ∈ {1, . . . ,L} and all configurations ω. Using this C1 in (48) we furtherestimate

P(E0

(HN

ω,L

)� E

)� P

(L∑

i=1

E0(ωi) − E0

L� s0

)� e−γ0L.

Here the last step is a large deviations bound, which is applicable with suitably chosen s0 > 0and γ0 > 0 due to the assumption (42). Note for this that E(ωi) − E0 are non-negative randomvariables which are strictly positive with positive probability. With this s0 in (48) if follows thate−γ0L � C(E − E0)

α , where α := s0/4C1. This completes the proof.

5. Concluding remarks

With the above results we have only started to touch the various possibilities for the low-energy asymptotics of the IDS in the random displacement model. There are several otherregimes which we have not considered yet:

(i) For one-dimensional random displacement models with non-symmetric distribution, in par-ticular the case μ = pδdmax + (1 − p)δ−dmax with p �= 1/2 we expect that the IDS might haveLifshits tails.

(ii) It would be most interesting to decide if the uniqueness of the minimizing periodicconfiguration established in Theorem 2.4(b) leads to Lifshits tails of the IDS at E0 for themulti-dimensional random displacement model with general (or suitable) distributions μ. Be-yond uniqueness of the minimizing configuration this would require to have quantitative resultson the probability that other configurations have ground state energy near E0.

In this context we mention the recent work of Klopp and Nakamura [11] on sign-indefiniteAnderson models, where some phenomena similar to those found by us for the random displace-ment model appear. In particular, they find that Lifshits tail as well as van Hove asymptoticsof the IDS at the bottom of the spectrum are both possible in their model, depending on thechoice of single-site potential and distribution of the random parameters. They have informed usabout work in preparation [12] which, when combined with the uniqueness result Theorem 2.4(b)above, should indeed lead to Lifshits-type asymptotics of the IDS for multi-dimensional randomdisplacement models as considered here. Their methods require that the distribution μ of thedisplacements has discrete support containing the corners of [−dmax, dmax]d .

(iii) We have used the non-overlap condition suppμ ⊂ [−dmax, dmax]d , dmax + r = 1/2,mostly for technical reasons. In particular, it is crucial for the Neumann-bracketing argumentsused in [2] and also Section 3 above. However, relaxing this condition will also lead to differentphenomena. We mention the recent work by Fukushima [8] who studies the random displace-ment model (1), (2) for positive q and displacements with unbounded distribution μ. In this caseit is easily seen that the almost sure spectrum is [0,∞), due to the presence of large empty re-gions in typical single-site configurations (while in our setting the spectral minimum would bestrictly positive). Under this condition Fukushima establishes Lifshits tails of the IDS at 0. An-other interesting task would be to look at intermediate cases, where suppμ is bounded but notsmall, allowing overlapping finite clusters of single-site potentials, but no large empty regions.

(iv) Under alternative (ii) all random configurations give the same ground state energy. Thisis an example of a random operator with a stable spectral boundary (as opposed to fluctuationboundaries). In other examples of this type, for a discussion see Sections 6B and 9 of [13], this


has been found to lead to van Hove behavior of the IDS, i.e. N(E) ∼ (E − E0)d/2 as for the

unperturbed Laplacian. We also expect this here.

Acknowledgments

M.L. would like to acknowledge partial support through NSF grant DMS-0600037. G.S. waspartially supported through NSF grand DMS-0653374.

References

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